Beverton–Holt model

The Beverton–Holt model is a classic discrete-time population model which gives the expected number n_t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,

n_{t+1} = \frac{R_0 n_t}{1+ n_t/M}.

Here R₀ is interpreted as the proliferation rate per generation and K = (R₀ − 1) M is the carrying capacity of the environment. The Beverton–Holt model was introduced in the context of fisheries by Beverton & Holt (1957). Subsequent work has derived the model under other assumptions such as contest competition (Brännström & Sumpter 2005) or within-year resource limited competition (Geritz & Kisdi 2004). The Beverton–Holt model can be generalized to include scramble competition (see the Ricker model, the Hassell model and the Maynard Smith–Slatkin model). It is also possible to include a parameter reflecting the spatial clustering of individuals (see Brännström & Sumpter 2005).

Despite being nonlinear, the model can be solved explicitly, since it is in fact an inhomogeneous linear equation in 1/n. The solution is

n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}.

Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is

\frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right),

and its solution is

N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}.

References

Beverton, R. J. H.; Holt, S. J. (1957), On the Dynamics of Exploited Fish Populations, Fishery Investigations Series II Volume XIX, Ministry of Agriculture, Fisheries and Food

Brännström, Åke; Sumpter, David J. T. (2005), "The role of competition and clustering in population dynamics" (PDF), Proc. R. Soc. B 272 (1576), pp. 2065–2072, doi:10.1098/rspb.2005.3185, PMC 1559893, PMID 16191618

Geritz, Stefan A. H.; Kisdi, Éva (2004), "On the mechanistic underpinning of discrete-time population models with complex dynamics", J. Theor. Biol. 228 (2), pp. 261–269, doi:10.1016/j.jtbi.2004.01.003, PMID 15094020

Ricker, W. E. (1954), "Stock and recruitment", J. Fisheries Res. Board Can. 11, pp. 559–623

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